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On global rigidity of transversely holomorphic Anosov flows

Mounib Abouanass

TL;DR

The paper addresses the global rigidity of transversely holomorphic Anosov flows, showing that when the strong unstable distribution has complex dimension 1, strong leaves acquire complete complex affine structures and the flow acts holomorphically on them; under topological transitivity, weak foliations gain transverse projective structures. The authors develop a framework based on holonomy, complex structures on leaves, and transverse (G,X)-structures, and prove the existence and uniqueness of affine structures on strong leaves via a jet/contracting-map approach, alongside transverse projective structures for weak foliations. In dimension five, these transverse structures enable a global developing map and holonomy representation, leading to a sharp classification: the flow is $C^{\infty}$-orbit equivalent to either the suspension of a hyperbolic automorphism of a complex torus or to the geodesic flow on the unit tangent bundle of a compact hyperbolic 3-manifold, after finite covers. Overall, the work extends Ghys-type rigidity into the holomorphic setting, delivering a robust method to classify transversely holomorphic Anosov flows via developing maps and holonomy representations.

Abstract

In this paper, we study transversely holomorphic flows, i.e. those whose holonomy pseudo-group is given by biholomorphic maps. We prove that for Anosov flows on smooth compact manifolds, the strong unstable (respectively, stable) distribution is integrable to complex manifolds, on which the flow acts holomorphically. Furthermore, assuming its complex dimension to be one, it is uniquely integrable to complex affine one-dimensional manifolds, each moreover affinely diffeomorphic to $\mathbb C$, on which the flow acts affinely. In this case, the weak stable (respectively, unstable) foliation is transversely holomorphic, and even transversely projective if the flow is assumed to be topologically transitive. By combining these facts in low dimensions, our main result is as follows : if a transversely holomorphic Anosov flow on a smooth compact five-dimensional manifold is topologically transitive, then it is either $C^\infty$-orbit equivalent to the suspension of a hyperbolic automorphism of a complex torus, or, up to finite covers, $C^\infty$-orbit equivalent to the geodesic flow of a compact hyperbolic manifold.

On global rigidity of transversely holomorphic Anosov flows

TL;DR

The paper addresses the global rigidity of transversely holomorphic Anosov flows, showing that when the strong unstable distribution has complex dimension 1, strong leaves acquire complete complex affine structures and the flow acts holomorphically on them; under topological transitivity, weak foliations gain transverse projective structures. The authors develop a framework based on holonomy, complex structures on leaves, and transverse (G,X)-structures, and prove the existence and uniqueness of affine structures on strong leaves via a jet/contracting-map approach, alongside transverse projective structures for weak foliations. In dimension five, these transverse structures enable a global developing map and holonomy representation, leading to a sharp classification: the flow is -orbit equivalent to either the suspension of a hyperbolic automorphism of a complex torus or to the geodesic flow on the unit tangent bundle of a compact hyperbolic 3-manifold, after finite covers. Overall, the work extends Ghys-type rigidity into the holomorphic setting, delivering a robust method to classify transversely holomorphic Anosov flows via developing maps and holonomy representations.

Abstract

In this paper, we study transversely holomorphic flows, i.e. those whose holonomy pseudo-group is given by biholomorphic maps. We prove that for Anosov flows on smooth compact manifolds, the strong unstable (respectively, stable) distribution is integrable to complex manifolds, on which the flow acts holomorphically. Furthermore, assuming its complex dimension to be one, it is uniquely integrable to complex affine one-dimensional manifolds, each moreover affinely diffeomorphic to , on which the flow acts affinely. In this case, the weak stable (respectively, unstable) foliation is transversely holomorphic, and even transversely projective if the flow is assumed to be topologically transitive. By combining these facts in low dimensions, our main result is as follows : if a transversely holomorphic Anosov flow on a smooth compact five-dimensional manifold is topologically transitive, then it is either -orbit equivalent to the suspension of a hyperbolic automorphism of a complex torus, or, up to finite covers, -orbit equivalent to the geodesic flow of a compact hyperbolic manifold.
Paper Structure (19 sections, 62 theorems, 120 equations)

This paper contains 19 sections, 62 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Reminders and definitions
  3. Foliations
  4. Holonomy
  5. Almost complex structure
  6. Transversely holomorphic foliations
  7. Hyperbolic set, Anosov flows
  8. Quasiconformality
  9. Fundamental examples of transversely holomorphic flows on smooth five-dimensional manifolds
  10. Suspension of a holomorphic diffeomorphism of a complex surface
  11. Geodesic flow on a hyperbolic three-dimensional manifold
  12. Sketch of proof of the results
  13. Complex structures of strong leaves and transverse structures of weak foliations
  14. The strong leaves are complex
  15. Transverse holomorphic structure of the weak foliations
  16. ...and 4 more sections

Key Result

Theorem 2.8

An almost complex structure $J$ on a smooth manifold $M$ is integrable if and only if its Nijenhuis tensor vanishes identically.

Theorems & Definitions (120)

  • Definition 2.2
  • Definition 2.3
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Definition 2.9
  • Proposition 2.10
  • proof
  • Definition 2.11
  • ...and 110 more